Question

state where the graph of the parabola is increasing and where it is decreasing.
$y = 4x^2 - 4x - 3$
state where the graph of the parabola is increasing.
x

Explanation:

Step1: Find the vertex of the parabola

For a quadratic function in the form \( y = ax^2 + bx + c \), the x - coordinate of the vertex is given by \( x = -\frac{b}{2a} \).
In the function \( y = 4x^2 - 4x - 3 \), \( a = 4 \) and \( b=-4 \).
Substitute \( a = 4 \) and \( b = - 4 \) into the formula for the x - coordinate of the vertex:
\( x=-\frac{-4}{2\times4}=\frac{4}{8}=\frac{1}{2} \)

Step2: Determine the direction of the parabola

Since the coefficient of \( x^2 \) ( \( a = 4 \)) is positive, the parabola opens upwards.

Step3: Determine where the function is increasing

For a parabola that opens upwards, the function is increasing to the right of the vertex (i.e., for \( x \) values greater than the x - coordinate of the vertex).
Since the x - coordinate of the vertex is \( \frac{1}{2} \), the function \( y = 4x^2-4x - 3 \) is increasing for \( x>\frac{1}{2} \).

Answer:

The graph of the parabola \( y = 4x^2 - 4x - 3 \) is increasing for \( x > \frac{1}{2} \).